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Mirrors > Home > MPE Home > Th. List > tgbtwntriv2 | Structured version Visualization version GIF version |
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgbtwntriv2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 768 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥)) | |
2 | tkgeom.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tkgeom.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
4 | tkgeom.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tkgeom.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | ad2antrr 723 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG) |
7 | tgbtwntriv2.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | 7 | ad2antrr 723 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃) |
9 | simplr 766 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃) | |
10 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵)) | |
11 | 2, 3, 4, 6, 8, 9, 8, 10 | axtgcgrid 28147 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥) |
12 | 11 | adantrl 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥) |
13 | 12 | oveq2d 7428 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥)) |
14 | 1, 13 | eleqtrrd 2835 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵)) |
15 | tgbtwntriv2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | 2, 3, 4, 5, 15, 7, 7, 7 | axtgsegcon 28148 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) |
17 | 14, 16 | r19.29a 3161 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 distcds 17213 TarskiGcstrkg 28111 Itvcitv 28117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-trkgc 28132 df-trkgcb 28134 df-trkg 28137 |
This theorem is referenced by: tgbtwncom 28172 tgbtwntriv1 28175 tgcolg 28238 legid 28271 hlid 28293 lnhl 28299 tglinerflx2 28318 mirreu3 28338 mirconn 28362 symquadlem 28373 outpasch 28439 hlpasch 28440 |
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